Question

Using the following equation, find the center and radius of the circle. You must show all work and calculations to receive credit.
x^2 - 4x + y^2 + 8y = -4

Answer 1

I looked around and I couldn't find any answer to any question related to this :/

Answer 2

An easy way to do this is by completing the square, do you know this method?

Answer 3

no... :/

Answer 4

Well let's first look at (x^2 - 4x), we need to get this into the cleaner (x + a)^2 form.

The method is to make it so that (x + a)^2 = (x^2 - 4x + b)

Answer 5

So we halve the coefficient of the second term (-4) and square it, 2^2 = 4. We add this new 4 to (x^2 - 4x) to make (x^2 - 4x + 4). Now the equation is factorable.

However, we can't just create the new 4 out of the dust. If we add 4 to one side of the equation, what must we do with the other?

Answer 6

so if we add it to the other we get (x^2-4x+4)=0?

Answer 7

not including the y terms

Answer 8

Yes. Now factor the quadratic.

Answer 9

I'm sorry I haven't done factoring for a year, how would I do that?

Answer 10

What two numbers multiply to positive 4, but add to negative 4?

Answer 11

-2 and -2?

Answer 12

Correct. Using that information, what two terms multiplied would give you x^2 - 4x + 4?

Answer 13

i have no idea

Answer 14

Remember when we're factoring, we're creating two factors. Those are things that multiply to form a product. Can you identify the product?

Answer 15

x^2 - 4x + 4?

Answer 16

Yes, and what format do you usually see factors in? HINT: ( ) * ( )

Answer 17

(x +/- #)(x +/- #)?

Answer 18

where # is a number

Answer 19

Right. Now pretend this is (x + 1)(x + 2), what would this look like if you multiplied it out?

Answer 20

x^2+2x+1x+2=x^2+3x+2

Answer 21

3*

Answer 22

the last number is 3

Answer 23

wait no its not whyd i doubt myself

Answer 24

*facepalm* sorry

Answer 25

You were right the first time. Anyway, the goal is to find two binomials (x + m)(x + n) that = x^2 - 4x + 4 when multiplied out.

Answer 26

ok so would it be (x-2)(x-2)?

Answer 27

Yes, and since they are the same, how else could you write that?

Answer 28

(x-2)^2

Answer 29

Perfect. Now that we've simplified that, can you tell me what the equation looks like now?

Answer 30

(x-2)^2 + (y^2 + 8y) = 0

Answer 31

Right. Now, the x part is finished. Try running through the same process with the y.

Answer 32

ok

Answer 33

so i halve 8 and get 4 then i square it and get 16 and add it to both sides right

Answer 34

Correct.

Answer 35

so (x+4)^2?

Answer 36

Yes, with a y instead of x. What does the equation look like now?

Answer 37

oh yeah (x+2)^2 + (y+4)^2 = 16

Answer 38

(x-2)^2, be careful not to switch up your signs

Answer 39

so the center is (-2, 4) and the radius is 16, correct (and thank you i wouldve forgot that)

Answer 40

Whoops, looks like you got a couple things confused.

Answer 41

i did? oops

Answer 42

(x – h)^2 + (y – k)^2 = r^2 is the equation for a circle. if r^2 is 16, then what should the radius be?

Answer 43

oh ok so the radius would be 4?

Answer 44

Right. 2nd thing: A circle of this equation is centered at (h, k). if the equation is (x – h)^2 + (y – k)^2 = r^2, then where should the center be?

Answer 45

oh so its (+2, -4) i got them switched oops

Answer 46

That's right. Good work.

Answer 47

thank you ^-^//

Answer 48

You're welcome :]